Seismic Risk Assessment Procedures for a System consisting of Distributed Facilities -Part three- Insurance Portfolio Analysis M. Achiwa & M. Sato Yasuda Risk Engineering Co., Ltd., Tokyo, Japan M. Mizutani Modern Engineering & Design Co., Ltd., Tokyo, Japan Keywords: insurance portfolio, seismic risk curve, computational time, seismic loss function, scenario earthquakes, PML ABSTRACT: The practical estimation procedures of the seismic risk curves for insurance portfolios are developed based on the seismic risk assessment procedures for a system consisting of distributed facilities, presented in the part one of this series of the study. Several improvements adjusted for the seismic portfolio analysis are introduced to reduce the computational time. As for the example to examine the applicability of the procedures, a hypothetical insurance portfolio including thousands of facilities located all over Japan is prepared and evaluated its risk curves. The seismic hazard used for the example calculation is modeled as presented in the part two of the series of the study. It is concluded that the proposed procedures are applicable although there are a little needs for further improvement. 1 INTRODUCTION An insurance portfolio means a set of policies retained by an insurance company. The estimation of payment for the entire portfolio is one of the most important issues for an insurance company. In general, when the numbers of polices is large the total amount of the payment for the portfolio converges to the expected value as the law of large numbers explains. However, as for the insurances of rare events such as earthquake damages, it is hardly believed that the expected value realize even for a large portfolio. Those insurances of rare events are so called cat losses by insurers. For the portfolio including cat losses, the probable maximum loss (PML) has been estimated as a part of the decision-making information for an insurance company. However the definitions of PMLs and/or methodologies to estimate the PML have not been fully established. Recently, instead of the PMLs the risk curves are becoming popular, as that will give better information to an insurance company, which indicates loss amount versus occurrence frequency. In general, a risk curve may be statistically estimated based on the payment records for a long period. As for very rare events like earthquake damages, it is not expectable to have enough records for this estimation of risk curves. For such cases, the seismic risk curves (SRCs) should be estimated introducing engineering procedures. In this paper, we propose a practical estimation procedure of seismic risk curve for an insurance portfolio. We prepared hypothetical seismic insurance portfolio consisting of thousands of building and facilities distributed all over Japan. The basic concept of the procedures is presented in the part one of this series of study, and seismic hazard model in and around Japan is as presented in the part two of the series. Several improvements of the procedures to reduce the computational efforts are introduced in this study. The sensitivity study to the size of the set of earthquakes is also conducted. 1
2 IMPROVEMENT OF THE PROCEDURES For the estimation of the seismic risk curve of the example portfolio that includes about 2,000 facilities under the seismic circumstance expressed by tens of thousands earthquakes, enormous computational effort should be required if we directly apply the procedures explained in the part one of this series of study. We introduced several improvements to the procedures adjusted for the insurance portfolio analysis, in order to reduce the computational efforts to estimate the seismic risk curve. 2.1 System model In this study, the insurance portfolio is considered as a system consisting of multiple facilities. For this system, the loss is the payment for the entire portfolio and is the summation of the payment for each individual facilities as there is no system interaction for the payment. Then we need not to develop a complicated event tree (ET) for the system. 2.2 ETs for facilities The example portfolio includes 2,000 facilities and it is not practical to model every facility. We categorize the facility into several groups considering the structures, functions and ground conditions. For each category, we developed ET model. 2.3 Variability of the damage states In the procedures presented in the part one of the study, the variability of the damage states is considered for individual facility. In the insurance portfolio case, there are lots of facilities to be damaged by one earthquake in most cases. Then the summation of the loss should converge to the expected value. Based on this reason, we neglect the variability of the damage states for each facility and just estimate the expected loss conditional to each earthquake. 3 ESTIMATION PROCEDURES WITH AN EXAMPLE In this chapter, the procedures adjusted to the estimation of risk curves of insurance portfolio are presented with an example. Firstly, the outline of the procedures is shown, then the each procedure is explained with the example. 3.1 Outline of the procedures The outline of the procedures is depicted in the Figure 3-1 and is as the followings: 1) Categorize all the buildings into several groups considering their ground conditions, structures and functions. In this example case 192 categories are prepared. 2) Model the seismic loss sequences of each category into event tree (ET), which includes ground liquefaction, structural failure and fire followings. 3) Assign loss amount to each end brunch of the ET adapting to the insurance policies. 4) Generate the seismic loss function (SLF) of each category by quantifying the ET for every ground motion level. Here, ground motion is expressed by an averaged spectral acceleration (ASA) on the rock surface and the effects of surface layers are included in each event tree. 5) Estimate ASAs on the rock surfaces of all the locations of the buildings by one scenario earthquake using an attenuation relationship. 6) Evaluate the conditional loss due to the scenario earthquake of each building using corresponding SLF and obtain the expected loss. 7) Sum up the expected losses of all the buildings in the system and evaluate the conditional portfolio loss to the scenario earthquake. 8) Carry out the procedures 5),6) and 7) for all the scenario earthquakes. 2
9) Analyze statistically the results obtained in 8) with the occurrence frequencies of corresponding scenario earthquakes, and generate the SRC of the system. 3.2 Insurance portfolio Information In this study, a hypothetical insurance portfolio consists of about 2,000 facilities distributed in all over Japan is prepared (See Figure 3-2). As for the portfolio information, several items are listed in Table 3-1. A few of these items are explained as follows: Category of Use means what category is fit to use of that facility, for example Office, Factory, Petrochemical plant, etc. Soil type is categorized into 16 types by characteristics of seismic motion s amplification and liquefaction potential. Fire hazard potential is categorized into 3 types by amounts of ignition sources and flammable materials. Seismic Hazard Information a set of Scenario Earthquake Insurance Portfolio Information a set of Facilities Scenario Earthquake (i) Location (j) Facility (j) Attenuation Relationship Ground Motion (i,j) Seismic Loss Function (j) Event Tree (j) Seismic Loss (i,j) Summation Portfolio Loss (i) Statistical Analysis Seismic Risk Curve of the portfolio Figure 3-1 Flowchart of the Procedures 3
Table 3-1 Information of Portfolio Items Location Structure Category of Use Soil Type Fire Hazard Potential Insurance Condition Detail Longitude, Latitude Reinforced concrete frame, Steel frame Office, Factory, Petrochemical Plant, etc. 16 categories Dangerous, Normal, None Replacement Value, Limit of Loss, Deductible, etc 3.3 Development of SLF (Seismic Loss Function) Figure 3-2 Hypothetical Insurance Portfolio 3.3.1 Event Tree Modeling Seismic events are classified as ground liquefaction, shock loss, fire following earthquake and so on. These sequential damage events are modeled by using event tree (ET) technique. Occurrence probability of each damage event is obtained as conditional probability of seismic ground motion via seismic fragility curve. Some common quantitative measures of seismic motion are peak ground acceleration (PGA), response acceleration, peak ground velocity (PGV), and so on. In this study, we selected spectral response acceleration as the ground motion parameter, which is concerned for natural period of the building. For each damage state, probability is calculated by multiplying the branch probabilities lying in the sequence. Damage ratio of each damage state is defined based on insurance policy. Each loss ratio is multiplied probability by the damage ratio of each damage state. The total loss ratio is the summation of loss ratio of all the damage states. 4
Peak Acceleration on bedrock 377.947 cm/s 2 Peak ground Acceleration 499.673 cm/s 2 Response Acceleration 742.7654 cm/s 2 Ground Liquefaction Potential 3 Liquefaction Shock Loss Probability Damage Ratio Loss Ratio Slight Damage 0.78 Severe Damage 0.22 Slight Damage 0.68 Moderate Damage 0.28 Severe Damage 0.04 0.5334 0.2161 0.0299 0.2206 * * * * Figure 3-3 Example of Event Tree Model 0 20 100 100 = = = = 0.00 4.32 2.99 22.06 Total Loss Ratio(%) 29.37 3.3.2 Seismic Fragility Curve Seismic fragility curves for liquefaction, shock loss and fire following earthquake and so on are defined as log-normal distribution (see Figure 3-4). Median value and variance depend on types of structure and industry for the objective facility. 3.3.3 Seismic Loss Function Seismic Loss Function (SLF) shows the conditional expected loss of a facility given the ground motion level, which is obtained by quantifying the ET of the facility (see Figure 3-5). The SLFs of all the categories are calculated in advance to reduce the computational time of the risk estimation. With the SLFs, we can avoid to quantify ETs of all the facilities for every earthquake. Probability 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Slight Da mage Moderate Damage Severe Damage 0 1,000 2,000 3,000 4,000 Response Acceleration (cm/s 2 ) Figure 3-4 Seismic Fragility Curve 5
100 80 Estimated Loss(%) 60 40 Subsoil 1 Subsoil 2 20 Subsoil 3 0 0 400 800 1200 1600 2000 Bedrock ASA(cm/s 2 ) As for the ground motion level, the spectral acceleration on the bedrock is preferred, which can represent the vibratory characteristics of the facility better than peak ground acceleration. ASA on bedrock estimated by attenuation equation has uncertainty of itself. In order to consider with this uncertainty, calculated ASA on bedrock is median value and the probability of this uncertainty is defined as log-normal distribution with standard deviation, 0.4. The loss ratio calculated as probability density function is averaged at each ASA on bedrock and median value of ASA on bedrock is translated to mean value. 3.4 Seismic Hazard Information The seismic hazard in and around Japan is modeled as the set of scenario earthquakes which is characterized by the location (rupture area), magnitude, and occurrence rate (Refer Part two). 3.5 Estimation Method 3.5.1 Scenario Earthquake (i) One scenario earthquake is selected in the set of scenarios prepared in advance. 3.5.2 Ground Motion Relationship Attenuation equation used in this study is Annaka Model, which is able to estimate spectral acceleration on bedrock. Log10A(t) = Cm(t)*M + Ch(t)*H Cd(t)*Log10D + Cc(t) D = R + 0.35*exp(0.65*M) A(t): spectral acceleration at period t M: magnitude H: depth at center of rupture area R: shortest distance from rupture area Figure 3-5 Example of Seismic Loss Function 6
3.5.3 Loss (i,j) Expected loss of each facility given the earthquake is estimated by corresponding SLF. 3.5.4 Portfolio loss (i) The portfolio loss given the earthquake is calculated by summing up the expected losses of all the facilities. Although the loss of individual facility has variability, the portfolio loss may converge to the expected value when the number of facilities is large. 3.5.5 Statistical Analyses After finishing the calculation for all scenarios of earthquake, portfolio loss with the occurrence frequency is sorted in descending order of loss. 3.5.6 Seismic Risk Curve Seismic Risk Curve shows exceedance probability of the portfolio loss. The results of 3.5.5 are converted to the SRC with the assumption of Poisson Occurrence of each earthquake. 4 ESTIMATION RESULTS OF THE EXAMPLE PORTFOLIO We estimated two seismic risk curves for the example portfolio. The difference between them is the number of scenario earthquakes used to represent the seismic circumstances. In the Figure 4-1, these curves are depicted. 4.1 Comparison of the two results Both estimation results are similar and seem to be adequate for the range where the exceedance probability is more than 1/500. Truncations appear at the point of 1/500 of exceeding probability in both cases. These truncations may be caused by the discrete expression of seismic hazard and by the neglecting of the variability of damage states for individual facilities. However for the practical purpose, 1/500 of exceeding probability is small enough. Estimated Loss Annual Exceeding Probability 1 0.1 0.01 0.001 0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 10,000 Scenarios 20,000 Scenarios 0.0001 Figure 4-1 Comparison of Seismic Risk Curve 7
Table 4-1 Comparisons of Estimated Loss and Risk Premium Scenario Loss of 99%ile Risk Premium 10,000 Scenario 36,254 3,203 20,000 Scenario 34,560 3,253 Table 4-1 shows the characteristic values obtained by these risk curves. The risk premium is an insurance term for the expected loss. Based on the comparison of these two results, 10,000 scenario earthquakes are considered large enough to represent the seismic hazard for the example portfolio. 4.2 Comparison of computational times The seismic risk curve estimation of a portfolio is considered to recognize large computational efforts. For the example portfolio, the computational times were 6.3 hours in 10,000scenario case and 34.2 hours in 20,000 case. The specification of personal computer used for calculation is as follows: CPU: Intel Pentium III 833MHz, Memories: 256MB, Hard Disk: 10GB, Operating system: Windows NT 4.0 (Japanese) The program code was written in Microsoft Visual Basic version 6.0 (SP3). Although these computational time look rather long, there are enough potential to reduce them by improving computer algorithm and by using more sophisticated computer languages. 5 CONCLUSIONS AND FUTURE ISSUES In this study a practical procedures to estimate a seismic risk curve for an insurance portfolio is presented. For an example, a hypothetical insurance portfolio consisting of about 2,000 facilities distributed in Japan was prepared and its risk curves were estimated. The estimation results seem to be adequate and the applicability of the procedures is presented. A sensitivity study on the size of the set of earthquakes represent the seismic circumstances in and around Japan was also conducted, and it is revealed that a set of about 10,000 scenario earthquakes can be used for the discrete representation of the seismic hazard of Japan for rather large portfolio. As for the future studies, the following issues may be listed. Although the procedures are applicable, the accuracy of the estimated risk curves is highly dependent on the accuracy of seismic loss function of each facility. The computational time for the estimation is still long. When, smaller probability than 1/500 is of concern, the practical procedure to account for the variability of loss of each facility should be developed. References 1)M. Mizutani, 1998. Basic Methodology of Seismic Risk Management (SRM) Procedures, Structural Safety and Reliability, Shiraishi, Shinozuka & Wen (eds) 1998 Balkema, Rotterdam, ISBN 90 5410 9785, pp.1581-1588 2)T.Annaka & Y.Nozawa, 1988. A Probabilistic model for Seismic Hazard Estimation in the Kanto District, Proc. 9th WCEE 8